EnVision 2.0 Fifth Grade Unit 4 - Linda Patterson

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FIFTH GRADEUnit 4Division of Whole Numbers and Decimals20 daysenVision 2.0 Topics 5-6Overarching Understandings:Using an understanding of place value, properties of operations, and operations with whole numbers,problems with decimals can be solved using addition, subtraction, multiplication, and division. Problems aresolved using multiple representations: concrete models, drawings, equations, and a written explanation of thereasoning used.Essential Questions: How can I use the four operations to solve real world problems? In what real life situations would you use decimals? What strategies can we use to efficiently solve problems? How can we use models and drawings to help solve problems with decimals? How can I use what I know about the operations with whole numbers to help solve problems withdecimals?Common Core State Standards:5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors,using strategies based on place value, the properties of operations, and/or the relationship betweenmultiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/orarea models.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings andstrategies based on place value, properties of operations, and/or the relationship between addition andsubtraction; relate the strategy to a written method and explain the reasoning used.Key d Materials:dicecolored markers/pensgrid paperSentence Frames:My estimate is .I used to solve the problem by .This model/drawing shows .place value chartscolored tilesNumber Talks: Number Talks are used to build number sense, develop fluency, and make sense of problems.Problem SolvingNumber StringsSDUSD Fifth Grade Unit 4 Overview1

FIFTH GRADEUnit 4Division of Whole Numbers and Decimals20 days19 lessons1 Assessment DaySuggested Order of LessonsObjective 1: Students will divide whole numbers using strategies based on place value, propertiesof operations, and the relationship between multiplication and division. (5.NBT.6 and 5.NBT.7)Day123456789SourceLesson TitleenVision 2.0 5-1 Use Patterns and Mental Mathto DivideenVision 2.0 5-2 Estimate Quotients with 2Digit DivisorsenVision 2.0 5-3 Use Models to Divide with 2Digit DivisorsenVision 2.0 5-4 Use Partial Quotients to DividePage NumberENV TE p. 239enVision 2.0 5-5 Divide by Multiples of 10ENV TE p. 263enVision 2.0 5-6 Use Estimation to Place theFirst Digit of the QuotientenVision 2.0 5-7 Divide by 2-Digit DivisorsENV TE p. 269GeorgiaAre These All 364/15enVision 2.0 5-8 Make Sense and PersevereNotesENV TE p. 245ENV TE p. 251ENV TE p. 257ENV TE p. 275Unit p. 9ENV TE p. 281Objective 2: Students will divide decimals using strategies based on place value, properties ofoperations, and the relationship between multiplication and division. (5.NBT.6 and 5.NBT.7)10111213enVision 2.0 6-1 Patterns for Dividing withDecimalsenVision 2.0 6-2 Estimate Decimal QuotientsENV TE p.301enVision 2.0 6-3 Use Models to Divide by a 1Digit Whole NumberenVision 2.0 6-4 Divide by a 1-Digit WholeNumberENV TE p.313SDUSD Fifth Grade Unit 4ENV TE p.307ENV TE p.3192

14151617181920Notes:enVision 2.0 6-5 Divide by a 2-Digit WholeENV TE p.325SDUSDDecimal Story ProblemsenVision 2.0 6-6 Use Number Sense to DivideDecimalsenVision 2.0 6-7 Divide by a DecimalUnit p. 14ENV TE p.331enVision 2.0 6-8 Continue to Divide by aDecimalenVision 2.0 6-9 Problem SolvingENV TE p.343ENV TE p.337ENV TE p.349AssessmentSDUSD Fifth Grade Unit 43

SDUSD Math Lesson MapNumber Talks15 minutesThe structure of math lessons should follow the Launch,Explore, Summarize format. This structure allowsstudents to explore mathematical concepts with rigor(fluency, concept development, and application) todevelop understanding in ways that make sense. Somerich tasks may take multiple days for students to explore.In these cases, each day should still follow the Launch,Explore, Summarize format.Number Talks are a chancefor students to come togetherto practice fluency and sharetheir mathematical thinking byengaging in conversationsand discussions aroundproblem solving and numbersense activities.LAUNCH (5–10 minutes)WHOLE GROUPEXPLORE (15–20 minutes)The teacher provides opportunities and support for students to develop conceptual understanding byproviding meaningful explorations and tasks that promote active student engagement.The teacher monitors the development of student understanding by conferring with students andasking students questions in order to understand and stimulate their thinking. The teacher uses thisinformation to plan for the Summarize and, if needed, to call the students together for a mid-Explore scaffoldto focus or propel student thinking.INDIVIDUAL, PAIRS, OR SMALL GROUPThe teacher provides opportunities to make public the learning that was accomplished by thestudents by sharing evidence of what was learned, and providing opportunities for students to analyze,compare, discuss, extend, connect, consolidate, and record thinking strategies. A summary of the learning isarticulated and connected to the purpose of the lesson.The students are actively engaged as a community of learners, discussing, justifying, and challengingvarious solutions to the Explore task. The students are able to articulate the learning/understanding of themathematical concept being taught either orally or in writing. Students can engage in this discussion whetheror not they have completed the task.WHOLE GROUPPRACTICE, REFLECT, and APPLY (10–15 minutes)This time is saved for after the Summarize so students can use what they have learned to access additionaltasks. The opportunities that teachers provide are responsive to student needs.The students may have the opportunity to: revise their work, reflect on their learning, show what theyknow with an exit slip, extend their learning with a similar or extension problem, or practice with centers orgames.The teacher confers with individual students or small groups.ASSESSMENTSUMMARIZE (15–20 minutes)FORMATIVEThe students are actively engaged in constructing meaning of the mathematical concept being taught.Students engage in private reasoning time before working with partners or groups. Students use multiplerepresentations to solve rich tasks and communicate their mathematical understanding.The teacher determines what students are learning and are struggling with by conferring with students and by examining student workthroughout the lesson. This formative assessment informs ongoing adjustments in the lesson and next steps for the class and each student.The students are actively engaged in a short task or discussion to activate prior knowledge in preparationof the Explore task. Students may be using tools and/or manipulatives to make sense of the mathematicalconcept.The students are actively engaged in showing their learning accomplishments related to the mathematical concept of the lesson.The teacher sets the stage for learning by ensuring the purpose and the rationale of the lesson are clearby connecting the purpose to prior learning, posing the problem(s), and introducing the Explore task forstudents. During this time the teacher is identifying the tools and materials available, reviewing academicvocabulary, and setting the expectations for the lesson.INDIVIDUAL, PAIRS, OR SMALL GROUPSDUSD Fifth Grade Unit 44

SDUSD Mathematics UnitsWe understand that for deep and sustainable change in mathematics to take place, teachers, students,and leaders must grapple with what the rich mathematics asked for by Common Core State StandardsMathematics looks like in the classroom, in pedagogical practice, in student work, in curriculum, and inassessments. It is our goal that teachers and site leaders work collaboratively toward a shared vision ofmath instruction that develops mathematically proficient students as defined by the CCSS-Mathematics. Itis our hope that these units provide a common instructional foundation for this collaboration.The SDUSD Mathematics Units are designed to support teachers and students as we shift from a moredirective style of teaching mathematics toward a more inquiry-based style. In problem-based learning,students develop the habits of mind and interaction of mathematicians through engaging in mathematicaldiscourse, connecting representations, asking genuine questions, and justifying and generalizing ideas.These mathematical habits reflect the shifts in pedagogy required to support the Common CoreStandards for Mathematical Practice.The SDUSD math units are compiled with multiple sources to ensure students have a variety ofmathematical experiences aligned to the CCSS. All lessons should follow the structure of Launch, Explore,and Summarize. The following document will guide teachers in planning for daily lessons, by helpingthem understand the structures of each of the sources.Structure for enVision 2.0 LessonsUse Step 1 Develop: Problem-Based Learning is the Launch, Explore, and Summarizefor every enVision 2.0 Lesson.Launch: (Before)Start with the Solve-and-Share problem. Pose the problem to the students making sure the problem isunderstood. This does not mean you explain how to do the problem, rather you ensure that studentsunderstand what the problem is about. Establish clear expectations as to whether students will workindividually, in pairs, or in small groups. This includes making sure students know whichrepresentations and tools they might be using or if they will have a choice of materials.Explore: (During)Students engage in solving the problem using a variety of strategies and tools. Use the suggested guidingquestions to check in briefly with students as needed, in order to understand and push student thinking.You may want to use the “Extension for Early Finishers” as needed.Summarize: (After)Select student work for the class to analyze and discuss. If needed, use the Sample Student Workprovided for each lesson in enVision 2.0.Practice, Reflect, Apply: (Select Problems from Workbook Pages, Reteach, Games, InterventionActivity)During this time, students may revise their work from the Explore time or you may use pieces of Step 2Develop: Visual Learning and Step 3 Assess and Differentiate. Note: The Quick-Check component is now aSDUSD Fifth Grade Unit 45

few select problems that are highlighted with a pink checkmark in the Teacher’s Edition. This timeprovides an excellent opportunity to pull small groups of students that may need additional support.Structure for Engage NY LessonsLaunch/Explore: (Concept Development)The Concept Development constitutes the major portion of instructional time when new learning isintroduced. During this time, the lessons move through a deliberate progression on material, fromconcrete to pictorial to abstract. Your word choice may be slightly different from that in the vignettes, andyou should use what works from the suggested talking points to meet your students’ needs.Summarize: (Student Debrief)The student debrief piece helps develop students’ metacognition by helping them make connectionsbetween parts of the lesson, concepts, strategies, and tools on their own. The goal is for students to seeand hear multiple perspectives from their classmates and mentally construct a multifaceted image of theconcepts being learned. Through questions that help make these connections explicit, and dialogue thatdirectly engages students in the Standards for Mathematical Practice, they articulate those observationsso the lesson’s objective becomes eminently clear to them.Practice, Reflect, Apply: (Problem Set/Exit Ticket)The Problem Set often includes fluency pertaining to the Concept Development, as well as conceptual andapplication word problems. The primary goal of the Problem Set is for students to apply the conceptualunderstandings learned during the lesson.Exit Tickets are quick assessments that contain specific questions to provide a quick glimpse of the day’smajor learning. The purpose of the Exit Ticket is twofold: to teach students to grow accustomed to beingindividually accountable for the work they have done, and to provide you with valuable evidence of theefficacy of that day’s work which is indispensible for planning purposes. This time provides an excellentopportunity to pull small groups of students that may need additional support.SDUSD Fifth Grade Unit 46

Structure for Georgia Standards LessonsThe Georgia Standards tasks have been included in the units to provide students opportunities for rich,engaging, real-world mathematical experiences. These tasks allow students to develop conceptualunderstanding over time and may take more than one math lesson to complete. The extra time for theselessons has been allotted for in the units. When planning for a Georgia Task, it is suggested that you startby doing the mathematics the students will be engaging in before presenting it to the students.Launch:You may need to activate prior knowledge for some of the tasks that will be presented by showingimages, letting students engage in partner talk about real-life situations, or using the suggested activityfrom the background knowledge component. Pose the task to the students making sure the task isunderstood. This does not mean that you explain how to do the problem, rather you ensure that studentsunderstand what the problem is about. You establish clear expectations as to whether students will workindividually, in pairs, or in small groups. This includes making sure students know which representationsand tools they might be using or if they will have a choice of materials.Explore:Students will engage in working on the task using a variety of strategies and tools. You may use theEssential Questions or Formative Assessment questions provided in the lesson as needed in order tounderstand and prompt student thinking.Summarize:Select student work for the class to analyze and discuss. Use partnerships and whole-class collaborativeconversations to help students make sense of each others’ work. The Formative Assessment questionsmay also be used during this time to facilitate the conversation.Practice, Reflect, Apply:At this time, provide students time to reflect and revise their work from the Explore after they haveengaged in the conversation in the Summarize portion of the lesson. This time provides an excellentopportunity to pull small groups of students that may need additional support.SDUSD Fifth Grade Unit 47

Common Core Approach to AssessmentAssessments provide ongoing opportunities for students to show their learning accomplishments inaddition to offering students a pathway to monitor their progress, celebrate successes, examine mistakes,uncover misconceptions, and engage in self-reflection and analysis. A central goal of assessments is tomake students aware of their strengths and weaknesses and to give them opportunities to try again, dobetter and, in doing so, enjoy the experience of seeing their hard work pay off as their skill andunderstanding increases. Furthermore, the data collected as a result of assessments represent invaluabletools in the hands of teachers and provides specific data about student understanding that can informinstructional decisions.For each Topic in enVision 2.0 the following assessments are available:In the Student Workbook:-Topic Assessment-Performance AssessmentOnline Teacher’s Edition:-Additional topic assessment Black-line Master-Additional performance assessment Black-line MasterOnline Student Assessment-Teacher can modify the number of items on an assessment-Teacher can rearrange order of problemsAll of the assessment items for enVision 2.0 are aligned to the types of problems students may encounteron state testing. We have found enVision 2.0 has an excessive amount of items suggested for each topic.To avoid over-assessing, we recommend that school sites work collaboratively in grade-level teams todetermine how to best use all the assessment resources available to evaluate student understanding andreduce the amount of items assessed. The SDUSD math units have grouped related topics together withina unit. Sites may choose to only give an assessment at the end of each unit, consisting of items frommultiple topics, rather than using multiple days to assess each topic individually.SDUSD Fifth Grade Unit 48

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Teacher EditionFifth Grade Mathematics Unit 1Constructing Task: Are These All 364 15?In this task, students analyze story problems that demonstrate three different division situations.STANDARDS FOR MATHEMATICAL CONTENTMCC5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends andtwo-digit divisors, using strategies based on place value, the properties of operations, and/or therelationship between multiplication and division. Illustrate and explain the calculation by usingequations, rectangular arrays, and/or area models.STANDARDS FOR MATHEMATICAL PRACTICESMP 1. Make sense of problems and persevere in solving them.SMP 2. Reason abstractly and quantitatively.SMP 3. Construct viable arguments and critique the reasoning of others.SMP 4. Model with mathematics.SMP 5. Use appropriate tools strategically.SMP 6. Attend to precision.SMP 7. Look for and make use of structure.SMP 8. Look for and express regularity in repeated reasoning.BACKGROUND KNOWLEDGESince third grade, students have worked with division through the use of partitioning wholenumbers, rectangular arrays area models and through the relationship of multiplication. Theyshould be able to apply these understandings of various division situations within this task.This standard references various strategies for division. Division problems can includeremainders. Even though this standard leads more towards computation, the connection tostory contexts is critical. Make sure students are exposed to problems where the divisor is thenumber of groups and where the divisor is the size of the groups. In fourth grade, students’experiences with division were limited to dividing by one-digit divisors. This standard extendsstudents’ prior experiences with strategies, illustrations, and explanations. When the two-digitdivisor is a “familiar” number, a student might decompose the dividend using place value.COMMON MISCONCEPTIONSStudents may not recognize the operation in a story problem situations. Students may translate adivision problem to a story problem that requires a different operation.ESSENTIAL QUESTIONS:SDUSD Fifth Grade Unit 4 Lesson 89

How can I use the situation in a story problem to determine the best operation to use?How can I effectively explain my mathematical thinking and reasoning to others?MATERIALS PaperPencilAccessible manipulativesGROUPINGindividual/partner taskTASK DESCRIPTION, DEVELOPMENT, AND DISCUSSIONCommentsThe three problems in this task represent situations where division can be used to solvedifferent kinds of problems.Problem A (measurement), creates a situation in which a given area must be divided todetermine the number of openings in the fencing. This situation promotes the strategy similarto the one listed below:Problem B calls for the partitioning of the money given by Old Mother Hubbard to her 15childrenIn Problem C, subtraction is used as a strategy to divide the given amounts. This is a lowlevel strategy, but it opportunity for students to connect their understanding of repeatedsubtraction to help develop a more efficient division strategy.Notice that students were not asked to actually solve any of these situations. The teachermay have students solve them either pictorially or using student invented strategies.Regardless, students should be required to explain their thinking.TASKStudents will follow the directions below from the “Are These All 364 15?” recordingsheet.You have been learning about many situations that can be solved with division. Eventhough the following problems all use the same numbers, think about whether eachdescribes a different type of division problem. After each problem explain why 364 15can or cannot be used to solve the problem.SDUSD Fifth Grade Unit 4 Lesson 810

Problem AThe new playground equipment was delivered to Anywhere Elementary School beforethe new fence was installed. Thomas Fencing Company arrived the next day with 364ftof fencing the school’s principal wanted an opening in the fence every 25 feet.According to the principal’s estimation the playground area would have about 15openings. The Thomas Fencing Company workers estimated 20 openings around theplayground. Who is correct? How do you know?Problem BOld Mother Hubbard found an old silver coin in her empty cupboard. She took it to theneighborhood coin collector and received 364 for the coin. With this increase inincome, Old Mother Hubbard was able to pay her children for the chores they completedduring the month. The 15 children inquired of their mother the amount of money eachwould receive. She was excited by the children’s inquiry and ran to the cupboard toretrieve beans to represent the money and Ziploc bags. Her kids were told to use thematerials to figure out the answer to their own question! What do you think theyfigured out and why?Problem CThe new poetry book by Mel Goldstein is 364 pages packed of humorous poems. LilyReader set a goal to read the entire book in 25 days. She planned to read 15 pages adays. With this plan, will she reach her goal? How do you know?On the back of this paper, write 3 of your own problems that can be solved using 252 12.FORMATIVE ASSESSMENT QUESTIONS How do you know which operation to use to solve the problem?What is happening to the whole or dividend within this situation?How many total parts does this situation involve?Does that amount make sense in this situation? Why or why not?How does this situation relate to multiplication?DIFFERENTIATIONExtensionStudents should be challenged to write problem situations that require a variety ofoperations and then solve them. Next, students can trade problems with a partner anddiscuss their solutions.Intervention Carefully screen the vocabulary to make sure that it is suitable for your students. Working in cooperative learning groups will support the student who is an Englishlanguage learner or for whom this task is challenging.SDUSD Fifth Grade Unit 4 Lesson 811

Name DateAre These All 364 15?You have been learning about many situations that can be solved withdivision. Even though the following problems all use the same numbers,think about whether each describes a different type of divisionproblem. After each problem explain why 364 15 can or cannot beused to solve the problem.Problem AThe new playground equipment was delivered to Anywhere Elementary School before the newfence was installed. Thomas Fencing Company arrived the next day with 364ft of fencing theschool’s principal wanted an opening in the fence every 25 feet. According to the principal’sestimation the playground area would have about 15 openings. The Thomas Fencing Companyworkers estimated 20 openings around the playground. Who is correct? How do you know?Problem BOld Mother Hubbard found an old silver coin in her empty cupboard. She took it to theneighborhood coin collector and received 364 for the coin. With this increase in income, OldMother Hubbard was able to pay her children for the chores they completed during themonth. The 15 children inquired of their mother the amount of money each would receive.She was excited by the children’s inquiry and ran to the cupboard to retrieve beans torepresent the money and Ziploc bags. Her kids were told to use the materials to figure outthe answer to their own question! What do you think they figured out and why?SDUSD Fifth Grade Unit 4 Lesson 812

Problem CThe new poetry book by Mel Goldstein is 364 pages packed of humorous poems. Lily Readerset a goal to read the entire book in 25 days. She planned to read 15 pages a days. With thisplan, will she reach her goal? How do you know?Write 2 of your own problems that can be solved using 252 12.1.2.SDUSD Fifth Grade Unit 4 Lesson 813

LESSON FOCUSMATERIALSLAUNCHFifth Grade Unit 4Lesson 15Decimal Story ProblemsBase-ten blocksSolving Problems1. Write the following problem on the board: Liz spent 12.72 on 6 bags ofchips. How much did she pay for each bag?2. Ask, “What do we need to find out?”3. Ask, “What strategy did you use to solve the problem?”4. Ask, “Did someone use a different strategy?”5. Have several students share their thinking on the board or overhead.6. Have students share an equation that matches this problem. (There maybe more than one)EXPLORESolving Story Problems1. Give students the problem solving worksheet.2. Students have three different number choices to use when solving theproblem. Tell students to choose a set of numbers to place in the problem.The numbers must be used in order in the problem. (ex: the first number inthe set goes in the first blank in the problem)3. Students will be given 5 minutes of Private Think Time (PTT) to engagein problem alone. Once PTT is over, have students partner talk to explainthe what and how of work, at end have them compare similarities anddifferences of work.4. Students use words, pictures and/or numbers to explain how they gottheir answer.5. Students choose another set of numbers to use in the problem as timeallows.SUMMARIZERefocus students on carpet sitting in a circle and pose following questions:Place a students work on the overhead, and pose question to otherstudents:How did make sense of what the problem was asking?How do you know?Note: This work could contain some errors. Ask students to analyze workfor what is correct, and where the error occurred. Have students partner talkbefore it goes to whole class so that every student has opportunity to share.OrExplain what did, and why they did it.Display work of student. Don’t let student who created work talk about thework, have other students make sense of what this student did.SDUSD Fifth Grade Unit 4 Lesson 1514

Name DateThe tennis team is having tennis rackets restrung. Thetotal cost is . What is the cost per racket for restringing?(3, 363)(10, 895)(8, 319.60)A developer owns acres of land. If he plans to useacres of the land for an entrance into a housingdevelopment and divide the remaining land intoacre lots, how many lots will he have?(18, 8, 0.5)(24, 1.2, 0.6)SDUSD Fifth Grade Unit 4 Lesson 15(48, 13.5, 2.3)15

4 enVision 2.0 5-4 Use Partial Quotients to Divide ENV TE p. 257 5 enVision 2.0 5-5 Divide by Multiples of 10 ENV TE p. 263 6 enVision 2.0 5-6 Use Estimation to Place the First Digit of the Quotient ENV TE p. 269 7 enVision 2.0 5-7 Divide by 2-Digit Divisors ENV TE p. 275 8 Georgia Are These All 364/15 Unit p. 9 9

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